Math 240A: Real Analysis, Fall 2011
Additional Exercise Problems
1. Let (X, ) be a metric space. Dene d : X X R by
d(x, y ) =
(x, y )
1 + (x, y )
x, y X.
Prove that (X, d) is also a metric space.
2. Let (X, ) be a metric space and E a closed subset of X .
Math 240A: Real Analysis, Fall 2012
Homework Assignment 6
Due Friday, November 9, 2012
Hints to Some Problems
Problem 2. Let f : R R be a Lebesgue integrable function. Prove that f = 0 m-a.e. if
and only if
f dm = 0
for all continuous function .
R
Hint. T
Math 240A: Real Analysis, Fall 2012
Additional Exercise Problems
1. Let (X, ) be a metric space. Dene d : X X R by
d(x, y ) =
(x, y )
1 + (x, y )
x, y X.
Prove that (X, d) is also a metric space.
2. Let (X, ) be a metric space and E a closed subset of X .
Math 240A: Real Analysis, Fall 2011
Homework Assignment 9
Due Friday, December 2, 2011
1. Exercise 22 on page 100.
2. Exercise 23 on page 100.
3. Exercise 24 on page 100.
4. Exercise 25 on page 100.
5. Let f L+ (Rn ). Prove that f dm is regular if and onl
Math 240A: Real Analysis, Fall 2011
Homework Assignment 8
Due Monday, November 28, 2011
1. Exercise 2 on page 88.
2. Exercise 3 on page 88.
3. Exercise 6 on page 88. [Note: the integral is over E .]
4. Exercise 8 on page 92.
5. Exercise 9 on page 92.
6. E
Math 240A: Real Analysis, Fall 2011
Homework Assignment 7
Hints
1. Exercise 23 on page 59.
Hint. See the hints in the text and also your class notes.
2. Exercise 38 on page 63.
Hint. For part (b) only. First, we have
fn gn f g = (fn f )(gn g ) + (fn f )g
Math 240A: Real Analysis, Fall 2011
Homework Assignment 7
Due Wednesday, November 16, 2011
1. Exercise 23 on page 59.
2. Exercise 38 on page 63.
3. Exercise 46 on page 68.
4. Exercise 48 on page 69.
5. Exercise 50 on page 69.
6. Exercise 53 on page 76.
7.
Math 240A: Real Analysis, Fall 2011
Homework Assignment 6
Due Monday, November 7, 2011
1. Let f : R R be a Lebesgue integrable function. Prove that f = 0 m-a.e. if and only
if
f dm = 0
for all continuous function .
R
2. Exercise 20 on page 59.
3. Exercise
Math 240A: Real Analysis, Fall 2011
Homework Assignment 5
Due Monday, October 31, 2011
1. Exercise 8 on page 48.
2. Exercise 9 on page 48.
3. Exercise 12 on page 52.
4. Exercise 14 on page 52.
5. Exercise 16 on page 52.
6. Exercise 18 on page 59.
7. Exerc
Math 240A: Real Analysis, Fall 2011
Homework Assignment 4
Due Friday, October 21, 2011
1. The Dirac measure concentrated on cfw_0 is a Borel measure on R. Find all the
increasing and right-continuous functions F : R R such that F = .
2. Let be a nite Bore
Math 240A: Real Analysis, Fall 2011
Homework Assignment 3
Due Friday, October 14, 2011
1. Does there exist an innite -algebra which has only countably many members?
2. Let (X, M, ) be a measure space. Assume that is -nite. Let E M with
(E ) = . Prove ther
Math 240A: Real Analysis, Fall 2011
Homework Assignment 2
Due Friday, October 7, 2011
1. Let X be a complete metric space and E a non-empty subset of X. Prove that E is
closed if and only if E is complete.
2. Let X be a metric space and E a compact subset
Math 240A: Real Analysis, Fall 2011
Homework Assignment 1
Due Friday, September 30, 2011
1. Let cfw_En be a sequence of sets. Dene
n=1
lim sup En = cfw_x : x En for innitely many n,
n
lim inf En = cfw_x : x En for all but nitely many n.
n
Prove that
lim
Math 240A: Real Analysis, Fall 2012
Additional Exercise Problems
1. Let (X, ) be a metric space. Dene d : X X R by
d(x, y ) =
(x, y )
1 + (x, y )
x, y X.
Prove that (X, d) is also a metric space.
2. Let (X, ) be a metric space and E a closed subset of X .